arXiv subjects

Alexey Fakhrutdinov

Publications and source records attributed to Alexey Fakhrutdinov.

2 recordsLinked to original sources

arXiv2026

Orthogonal partition by four hyperplanes

We study orthogonal $(4,3)$-equipartitions of finite measures absolutely continuous with respect to Lebesgue measure: every three of four mutually orthogonal affine hyperplanes must divide each measure into eight equal parts. For every number $m$ of measures, we determine the exact dimension threshold of Musin's polynomial criterion and express it in terms of the binary expansion of $m$. The threshold coincides with the geometric lower bound precisely when $m=2^j-1$. Consequently, $Δ^*(2^j-1,4,3)=7\cdot2^{j-1}-2$ for every $j\geq1$.

math.MG↗

arXiv2026

Algorithms for orthogonal partitioning into four parts

The famous pancake theorem states that for every finite set $X$ in the plane, there exist two orthogonal lines that divide $X$ into four equal parts. We propose an algorithm whose running time is linear in the number of points in $X$ and prove that this complexity is optimal. We also consider generalizations of the pancake theorem and show that orthogonal hyperplanes can be found in polynomial time.

math.CO↗